Galileo's Paradox
in [Math]
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From: David Marcus on 10 Dec 2006 14:29 Tony Orlow wrote: > David Marcus wrote: > > Tony Orlow wrote: > >> David Marcus wrote: > >>> Tony Orlow wrote: > > > >>>> Why do you have a problem with the mere suggestion of an infinite value? > >>> I don't have a problem with infinite values. However, you have to do > >>> more than merely say "positive infinite n". Assuming your positive > >>> infinite things aren't the same as something that we already know about > >>> (in which case, you should just say so), you either need to give a > >>> construction of these positive infinite things or you need to specify > >>> their properties. You haven't done either. For example, how many of > >>> these things are there? How do they relate to each other? How do they > >>> interact with the natural numbers? Are the operations of addition and > >>> multiplication defined for them? > >> Well, I have been through much of that regarding such specific language > >> approaches as the T-riffic digital numbers, but that's not necessary for > >> this purpose. It suffices to say that, if a statement is proved true for > >> all n greater than some finite k, that that also includes any postulated > >> infinite values of n, since they are greater than any finite k. I don't > >> need to construct these numbers. Consider them axiomatically declared. > > > > Then list the axioms for them. > > (sigh) > > infinite(x) <-> A yeR x>y Is that your only axiom? If so, then state your first theorem about them and give the proof. -- David Marcus
From: Mike Kelly on 10 Dec 2006 14:33 David Marcus wrote: > Tony Orlow wrote: > > David Marcus wrote: > > > Tony Orlow wrote: > > >> David Marcus wrote: > > >>> Tony Orlow wrote: > > > > > >>>> Why do you have a problem with the mere suggestion of an infinite value? > > >>> I don't have a problem with infinite values. However, you have to do > > >>> more than merely say "positive infinite n". Assuming your positive > > >>> infinite things aren't the same as something that we already know about > > >>> (in which case, you should just say so), you either need to give a > > >>> construction of these positive infinite things or you need to specify > > >>> their properties. You haven't done either. For example, how many of > > >>> these things are there? How do they relate to each other? How do they > > >>> interact with the natural numbers? Are the operations of addition and > > >>> multiplication defined for them? > > >> Well, I have been through much of that regarding such specific language > > >> approaches as the T-riffic digital numbers, but that's not necessary for > > >> this purpose. It suffices to say that, if a statement is proved true for > > >> all n greater than some finite k, that that also includes any postulated > > >> infinite values of n, since they are greater than any finite k. I don't > > >> need to construct these numbers. Consider them axiomatically declared. > > > > > > Then list the axioms for them. > > > > (sigh) > > > > infinite(x) <-> A yeR x>y > > Is that your only axiom? If so, then state your first theorem about them > and give the proof. The only thing I can come up with is "There are no infinite numbers". Am I close? -- mike.
From: Tony Orlow on 10 Dec 2006 14:52 David Marcus wrote: > Tony Orlow wrote: >> David Marcus wrote: >>> Tony Orlow wrote: >>>> David Marcus wrote: >>>>> Tony Orlow wrote: >>>>>> Why do you have a problem with the mere suggestion of an infinite value? >>>>> I don't have a problem with infinite values. However, you have to do >>>>> more than merely say "positive infinite n". Assuming your positive >>>>> infinite things aren't the same as something that we already know about >>>>> (in which case, you should just say so), you either need to give a >>>>> construction of these positive infinite things or you need to specify >>>>> their properties. You haven't done either. For example, how many of >>>>> these things are there? How do they relate to each other? How do they >>>>> interact with the natural numbers? Are the operations of addition and >>>>> multiplication defined for them? >>>> Well, I have been through much of that regarding such specific language >>>> approaches as the T-riffic digital numbers, but that's not necessary for >>>> this purpose. It suffices to say that, if a statement is proved true for >>>> all n greater than some finite k, that that also includes any postulated >>>> infinite values of n, since they are greater than any finite k. I don't >>>> need to construct these numbers. Consider them axiomatically declared. >>> Then list the axioms for them. >> (sigh) >> >> infinite(x) <-> A yeR x>y > > Is that your only axiom? If so, then state your first theorem about them > and give the proof. > That's the only one necessary for what defining a positive infinite n. A whole array of theorems pop forth from infinite-case induction and IFR, such as that the size of the even naturals is half that of the naturals. That's a no-brainer. Go back to where I first answered your question at length, and read again, at length. It's not transfinitology, but it's also not nonsense. Allow me to add another: |{ x| yeR and 0<(x-y)<=1}|=Big'un. That's the unit infinity. We map the hypernaturals in (0,Big'un] to the reals in (0,1] using f(n)=n/Big'un. Using the inverse function, g(x)=x*Big'un, over the range (0,Big'un], we get that there are Big'un^2 hyperreals, Big'un for each interval. This corresponds to the 2D array comprising the hyperrationals, if one replaces the redundant fractions with the equinumerous irrationals. :) TOny
From: David Marcus on 10 Dec 2006 15:44 Tony Orlow wrote: > David Marcus wrote: > > Tony Orlow wrote: > >> David Marcus wrote: > >>> Tony Orlow wrote: > >>>> Well, I have been through much of that regarding such specific language > >>>> approaches as the T-riffic digital numbers, but that's not necessary for > >>>> this purpose. It suffices to say that, if a statement is proved true for > >>>> all n greater than some finite k, that that also includes any postulated > >>>> infinite values of n, since they are greater than any finite k. I don't > >>>> need to construct these numbers. Consider them axiomatically declared. > >>> Then list the axioms for them. > >> (sigh) > >> > >> infinite(x) <-> A yeR x>y > > > > Is that your only axiom? If so, then state your first theorem about them > > and give the proof. > > > > That's the only one necessary for what defining a positive infinite n. A > whole array of theorems pop forth from infinite-case induction and IFR, > such as that the size of the even naturals is half that of the naturals. > That's a no-brainer. Go back to where I first answered your question at > length, and read again, at length. It's not transfinitology, but it's > also not nonsense. > > Allow me to add another: > > |{ x| yeR and 0<(x-y)<=1}|=Big'un. > > That's the unit infinity. > > We map the hypernaturals in (0,Big'un] to the reals in (0,1] using > f(n)=n/Big'un. Using the inverse function, g(x)=x*Big'un, over the range > (0,Big'un], we get that there are Big'un^2 hyperreals, Big'un for each > interval. This corresponds to the 2D array comprising the > hyperrationals, if one replaces the redundant fractions with the > equinumerous irrationals. :) OK. That's not the way the game is played. First you state all your definitions and axioms, then we see what theorems you can prove. Care to play again? -- David Marcus
From: David Marcus on 10 Dec 2006 15:47
Mike Kelly wrote: > David Marcus wrote: > > Tony Orlow wrote: > > > David Marcus wrote: > > > > Tony Orlow wrote: > > > >> Well, I have been through much of that regarding such specific language > > > >> approaches as the T-riffic digital numbers, but that's not necessary for > > > >> this purpose. It suffices to say that, if a statement is proved true for > > > >> all n greater than some finite k, that that also includes any postulated > > > >> infinite values of n, since they are greater than any finite k. I don't > > > >> need to construct these numbers. Consider them axiomatically declared. > > > > > > > > Then list the axioms for them. > > > > > > (sigh) > > > > > > infinite(x) <-> A yeR x>y > > > > Is that your only axiom? If so, then state your first theorem about them > > and give the proof. > > The only thing I can come up with is "There are no infinite numbers". > Am I close? You are about as close as I am. I can prove "A xeR (not infinite(x))". Doesn't seem like a very interesting theory, so far. -- David Marcus |